The speed that a sailboat is capable of sailing is...

The speed that a sailboat is capable of sailing is determined by three factors: its total length $L$, the surface area $A$ of its sails, and its displacement $V$ (the volume of water it displaces). In general, a sailboat is capable of greater speed if it is longer, has a larger sail area, or displaces less water. To make sailing races fair, only boats in the same "class" can qualify to race together. For a certain race, a boat is considered to qualify if $$0.30 L+0.38 A^{1 / 2}-3 V^{1 / 3} \leq 16$$ where $L$ is measured in feet, $A$ in square feet, and $V$ in cubic feet. Use this inequality to answer the following questions. a. A sailboat has length $60 \mathrm{ft}$, sail area $3400 \mathrm{ft}^{2},$ and displacement $650 \mathrm{ft}^{3}$. Does this boat qualify for the race? b. A sailboat has length $65 \mathrm{ft}$ and displaces $600 \mathrm{ft}^{3}$. What is the largest possible sail area that could be used and still allow the boat to qualify for this race?

The speed that a sailboat is capable of sailing is determined by three factors: its total length $L$, the surface area $A$ of its sails, and its displacement $V$ (the volume of water it displaces). In general, a sailboat is capable of greater speed if it is longer, has a larger sail area, or displaces less water. To make sailing races fair, only boats in the same "class" can qualify to race together. For a certain race, a boat is considered to qualify if $$0.30 L+0.38 A^{1 / 2}-3 V^{1 / 3} \leq 16$$ where $L$ is measured in feet, $A$ in square feet, and $V$ in cubic feet. Use this inequality to answer the following questions.
a. A sailboat has length $60 \mathrm{ft}$, sail area $3400 \mathrm{ft}^{2},$ and displacement $650 \mathrm{ft}^{3}$. Does this boat qualify for the race?
b. A sailboat has length $65 \mathrm{ft}$ and displaces $600 \mathrm{ft}^{3}$. What is the largest possible sail area that could be used and still allow the boat to qualify for this race?

Key Concepts

Exponentiation and Radical Operations

The problem makes use of square roots and cube roots, which are examples of non-linear exponentiation operations. Understanding how to manipulate expressions involving exponents—such as converting a square root to an exponent of 1/2, or a cube root to 1/3—is crucial. These operations allow one to treat the geometric aspects of physical quantities (like area and volume) appropriately within algebraic formulas.

Solving for an Unknown in an Inequality

Once an inequality is set up with some known values, it can be manipulated to solve for the unknown variable. This involves isolating the unknown term (in this case, a function of the sail area) and solving the inequality, often including steps such as taking square or cube roots, in order to find the maximum or minimum allowable values that comply with the given condition.

Dimensional Consistency

This concept refers to the practice of ensuring that all terms in an equation or inequality are consistent in terms of their units of measure. In the given problem, different physical quantities (length, area, and volume) measured in different units are combined linearly after being adjusted by coefficients. Recognizing and maintaining dimensional consistency helps in constructing valid mathematical models that accurately represent physical realities.

Algebraic Inequalities

This concept involves formulating and working with inequalities to determine whether certain conditions are met. In the context of the problem, an inequality is used as a qualifying criterion for sailboats, where specific weighted contributions from various parameters (such as length, sail area, and displacement) are compared against a threshold value. Understanding how to evaluate these expressions is key to determining compliance with the rule.

Substitution and Evaluation

Substitution is the process of replacing variables with given numerical values in an expression or an inequality. In problems like this, each boat’s specific measurements are substituted into the inequality, allowing for a concrete calculation that determines qualification status. This technique is foundational for applying abstract mathematical conditions to real-world scenarios.

Transcript

00:01 In part a of this question, consider expression to qualify or raise about 0 .30l plus 0 .38 to the power multiply by a to the power 1 divided by 2 minus 3b to the power 1 divided by 3 is less than equal to 16.

00:55 Now in above expression, l is measured in fit, a is measured in fit square and b is measured in fit cube.

01:03 So to solve the above expression, we substitute l equals 60 feet, a equals 3 ,000.

01:10 4 .0 feet square and b equals 650 cube.

01:16 So we get 0 .30 multiplied by 60 plus 0 .38 multiplied by 340 to the power 1 divided by 2 minus 3 multiplied by 650 to the power 1 divided by 2 is less than.

01:43 Equal to 16 now on simplifying this we will get 18 plus 0 .38 multiply by 50 0 .30 minus 3 multiply by 8 .47 is less than equal to 16 now further solve the above equation, we get 40 .154 minus 25 .41 is less than equal to 16.

02:37 From here we will get 14 .744 is less than equal to 16.

02:47 So, thus, sailboat of l equal 60 feet, a equal 3 ,400 feet square, and v equal 650 feet cube satisfy above consider consumption.

03:01 Hence it concludes that sailboat qualifies for the race.

03:28 Again in part v of this question consider expression that to qualify for race a boat as 0 .30l plus 0 .38 multiplied by a to the power 1 divided by 2 minus 3 v.

04:13 V.

04:15 2 is less than equal to 16.

04:21 Now here we substitute l equals 65, v.

04:27 Equal 600 feet to find the sailboat area so we get 0 .30 multiplied by 65 plus 0 .38 multiplied by a to the power 1 divided by 2 minus 3 multiply by 600 to the power 1 divided by 3 is less than equal to 16 from here we will get 19 .5 plus 0 .38 multiply by a 2d power 1 divided by 2 minus 3 multiply by 8 .3 is less than equal to 16 now on solve the equation further we will get 0 .38a to the power 1 divided by 2 minus 5 .8 is less than equal to 16...

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